2.5 Just in case ...

Let be the given function. Of course, this is the equation of a parabolid with its tip - or minimum - at the origin. However, let be the constraint equation, meaning that we don't search for the global minimum but for the minimum along the line . There are two ways to go about it. The simple but inconvenient way is to substitute in , thus rendering a function of only. Equating the derivative to zero we find the locus of the conditional minimum, and . The process of substitution is, in general, tedious.

A more elegant method is this: defining a (undetermined) Lagrange
multiplier ,
find the minimum of the function
according to

(2.53) | |||

(2.54) |

Eliminating we find the solution without substituting anything. In our case

(2.55) | |||

(2.56) |

, and from : .

2005-01-25